Convex solutions of certain elliptic equations have constant rank Hessians
In recent years there have been several results on the convexity of solutions to certain elliptic equations, e.g. \textit{N. Korevaar} [Indiana Univ. Math. J. 32, 603-614 (1983; Zbl 0481.35024)], \textit{A. Kennington} [ibid. 34, 687-704 (1985; Zbl 0549.35025)] and the reviewer [Z. Angew. Math. Mech. 64, No.5, T 364-T 366 (1984; Zbl 0581.73006); Math. Methods Appl. Sci. 8, 93-101 (1986; Zbl 0616.35006); Commun. Partial Differ. Equations 10, 1213-1225 (1985; Zbl 0587.35026)]. In contrast to these results the authors of the paper under review assume that u is convex and solves \(\Delta u=f(u,\nabla u)>0\) in \(\Omega \subset {\mathbb{R}}^ n\). The main result is the following. If 1/f(\(\cdot,\nabla u)\) is convex in u, then the Hessian H of u has constant rank in \(\Omega\). This implies in particular, that H has full rank in \(\Omega\) if it has full rank in a single point of \(\Omega\). The proof is a nontrivial extension of a former result of \textit{L. Caffarelli} and \textit{A. Friedman} [Duke Math. J. 52, 431-456 (1985; Zbl 0599.35065)]. The paper contains also applications of the main result.
scientific article; zbMATH DE number 4013141